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Maarten Bakker, Florent Gimbert, Thomas Geay, Clément Misset, Sébastien
Zanker, Alain Recking
To cite this version:
Maarten Bakker, Florent Gimbert, Thomas Geay, Clément Misset, Sébastien Zanker, et al.. Field
Application and Validation of a Seismic Bedload Transport Model.
Journal of Geophysical
Re-search:
Earth Surface, American Geophysical Union/Wiley, 2020, 125 (5), pp.e2019JF005416.
Maarten Bakker1,2 , Florent Gimbert2 , Thomas Geay3,4 , Clément Misset1 , Sébastien Zanker5, and Alain Recking1
1University Grenoble Alpes, INRAE, ETNA, Grenoble, France,2University Grenoble Alpes, CNRS, IRD, Institute for Geosciences and Environmental Research (IGE), Grenoble, France,3BURGEAP, Grenoble, France,4University Grenoble Alpes, CNRS, INP, GIPSA‐lab, Grenoble, France,5Électricité de France, DTG division, Grenoble, France
Abstract
Bedload transport drives morphological changes in gravel‐bed streams and sediment transfer in catchments. The large impact forces associated with bedload motion and its highly dynamicspatiotemporal nature make it difficult to monitor bedload transport in the field. In this study, we revise a physically‐based model of bedload‐induced seismic ground motion proposed by Tsai et al. (2012) and apply it to invert bedloadflux from seismic measurements alongside an Alpine stream. First, we constrain the seismic response of a braided river reach with a simple active experiment using a series of large‐rock impacts. This allows the characterization of surface wave propagation and attenuation with distance from the impact source. Second, we distinguish bedload‐generated ground vibrations from those caused by turbulentflow using frequency‐based scaling relationships between seismic power and discharge. Finally, absolute bedload transport rates are quantified from seismic measurements using inverse modeling based on a simplified formulation of bedload particle motion. The results are verified with a large data set of bedload samples, demonstrating that seismic measurements can provide an indirect measure for bedload flux with uncertainties within a factor of 5±1for instantaneous measurements (between 0.01 and 1 kg/m/s).
Larger deviations may be caused by uncertainties in the contribution of turbulentflow effects, particle impact velocity, and especially particle size that may vary with sediment supply andflow conditions. When constraining these uncertainties, instream sediment transport measurements are no longer necessarily required and seismic monitoring may provide an accurate and continuous means to investigate bedload dynamics in gravel‐bed streams.
1. Introduction
Sediment transport in gravel‐bed rivers is characterized by highly dynamic bedload processes that lead to largelyfluctuating transport rates (Gomez et al., 1989; Hoey, 1992). Quantifying bedload transport rates in both space and time is critical for the understanding of sediment transfer and coupled morphodynamics in river systems (Ashmore, 1991; Ashmore & Church, 1998; Williams et al., 2015), particularly in Alpine environments with high energy and sediment availability (e.g., Lane et al., 1996). The storage and/or release of riverbed material may affect riparian habitats and land usage through riverbed aggradation/degradation (Bakker et al., 2018) and lateral channel dynamics (Church, 2006). More directly, bedload impacts pose a direct hazard not only to infrastructure, including bridges and bank protection (Badoux et al., 2014), but also to instream measurement equipment that may be subject to damage or failure, particularly during high‐magnitude events. As such, there are only a limited number of sites with continuous bedload measure-ments (e.g., Mizuyama et al., 2010; Rickenmann et al., 2014). In order to acquire the necessaryfield data and to overcome practical limitations of direct measurements, seismic techniques have been proposed to provide an indirect measure of bedload transport.
Thefirst evidence that bedload transport may be detected in seismic data was provided by Govi et al. (1993). Some 15–20 years later, more extensive seismic measurements were made and explored for the frequency response of processes that generate seismic ground vibrations, most notably bedload transport andflow turbulence (Burtin et al., 2008; Burtin et al., 2011; Schmandt et al., 2013). The fundamental basis for the quantitative applications is provided by Tsai et al. (2012), who developed a physically‐based model and invert bedload transport from the seismic response it generates, and Gimbert et al. (2014), who complemen-ted this work with the modeling of turbulentflow that drives bedload transport.
© 2020. The Authors.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Key Points:
• Bedload fluxes are derived independently from bank‐side seismic measurements with an uncertainty factor of 5±1for
instantaneous values
• Simple experiments with large‐rock impacts are used to calibrate seismic response and provide an alternative to instream bedload sampling • Scaling seismic power with
discharge allows bedload‐generated vibrations to be isolated and gives insight into bedload temporal dynamics Supporting Information: • Supporting Information S1 Correspondence to: M. Bakker, maartenbakker@yahoo.com Citation:
Bakker, M., Gimbert, F., Geay, T., Misset, C., Zanker, S., & Recking, A. (2020). Field application and validation of a seismic bedload transport model. Journal of Geophysical Research: Earth Surface, 125, e2019JF005416. https://doi.org/10.1029/2019JF005416
Received 28 OCT 2019 Accepted 25 MAR 2020
The bedload model by Tsai et al. (2012) has been verified to be applicable for the quantification of low bed-load transport rates in controlledflume‐based experiments (Gimbert et al., 2019). However, the model has yet to be independently tested in thefield. For those studies where field observations are available, notably streams equipped with continuously monitoring Swiss plate geophone systems, empirical regression techni-ques (Roth et al., 2016; Roth et al., 2017) and Monte Carlo‐based modeling approaches (Dietze et al., 2019) were employed to quantify seismic response. However, the quantitative link with physical constraints was not investigated. Where no directfield observations were available or possible, a physically‐based approach was used to study transport during high‐flow events associated with typhoons (Chao et al., 2015), a monsoon season (Burtin et al., 2008; Tsai et al., 2012), and a glacial outburstflood (Cook et al., 2018). In this study, we aim to reconcile and combine these approaches by assessing bedload transport rates inverted from seismic measurements with independently sampled bedloadfluxes.
The goal of this paper is to evaluate how well bedload transport in a typical Alpine gravel‐bed stream can be constrained and quantified with seismic techniques. We assess the application of the theoretical seismic bed-load transport model of Tsai et al. (2012) in afield setting, a braided reach of the Séveraisse River in the French Alps. A two‐step approach is followed to validate the model. First, we characterize the seismic response of the riverbed through a simple active experiment using a series of large‐rock impacts. Second, we use this to quantify bedloadflux from seismic records using a physically‐based bedload saltation model. We compare the invertedfluxes to an extensive set of bedload samples at this site (Misset et al., 2020), demonstrating that a relatively simple seismic approach can provide a viablefield technique to continuously monitor bedload transport processes.
2. Study Area and Data Availability
The Séveraisse River is a natural Alpine stream thatflows from the Ecrins Massif in the French Alps. It has a catchment area of ~130 km2with the largestflows generated during spring snow melt and summer thunder-storms. We studied a braided river reach approximately 600 m long and up to 90 m wide (Figure 1) that lies at an elevation of just above 1,000 m a.s.l., near the village of Villar‐Loubière. The riverbed has a gentle gradient of ~1% and is largely composed of gravel at the surface and includesfiner material in the subsurface (Misset et al., 2019). The particle‐size distribution of the riverbed in the braided reach was determined by Misset et al. (2020) and is shown in Figure 2.
At the bridge just downstream of the braided reach (Figure 1), the electricity company Électricité de France (EDF) operates a gauging station that provides stage and discharge measurements at 10‐min resolution. The
Figure 1. Aerial image (courtesy of Esri‐World Imagery) of the studied reach of the Séveraisse River, France, with latitude and longitude coordinates on the x and y axes. Geophones locations L1, L2, R1, and R2 are shown, which were used for seismic characterization based on large‐rock impacts along cross sections between the geophones on opposite banks. At the downstream end of the reach, bedload sampling was performed and bedload transport is quantified from measurements at geophone B1. Inset image of geophone (right), logger (center), and GPS (left bottom).
channel here is straight and laterally confined, 12 m wide, and the riverbed is covered with cobbles that have an intermediate diameter D50= 11 cm and D84= 30 cm as derived from Wolman counts (particles sampled
at afixed interval, e.g., every 1 m, along a transect) by Misset et al. (2020). Misset et al. (2020) also performed an extensive bedload sampling campaign from the bridge (Figure 2; inset). Data are available from 60 cross‐sectional averaged bedload samples that were taken during a range of flow conditions (9–25 m3
/s) in the period 23 April to 26 June 2018. For 16 of these samples (240 kg in total), they also provide a particle‐size distribution by weight (Figure 2). In this study, we consider the measured bedload transport flux for the particle size range 1.6–91 mm. Both finer and coarser particles might not be efficiently captured dur-ing sampldur-ing due to the size of the mesh (0.5 mm) and mouth of the sampler (20.7 × 12 cm). Whereas the sand fraction has a negligible contribution to the seismic signal (Tsai et al., 2012), the load of the coarse frac-tion (>91 mm) may be estimated through extrapolafrac-tion using the particle‐size distribufrac-tion from mobile braided river bars (Parker & Klingeman, 1982) and accounts for 22% of the total load based on the data of Misset et al. (2020). Following Kellerhals and Bray (1971), particle‐size distributions found using volume by weight (sampled bedload of 91 mm and smaller) and grid by number (Wolman counts of riverbed) approaches are comparable, which is indeed confirmed in Figure 2.
3. Methods
3.1. Seismic Field Setup and Processing
In the period May–October 2018, five 3‐component Sensor PE‐6/B geophones registered ground motion at a distance of 25–50 m from the channel edge (Figure 1). Four geophones were placed along the braided reach and one besides the bridge where bedload sampling was performed. They were installed ~30 cm in the ground, oriented north and leveled horizontally. We study Rayleigh surface waves using measurements of the vertical component. The geophones allow seismic measurements over a frequency range from 4.5 Hz to >100 Hz, which is expected to be suitable for the observation of bedload transport at the considered dis-tance (Burtin et al., 2011; Gimbert et al., 2014). Data from the geophones were recorded with a frequency of 400 Hz on a DiGOS DATA‐CUBE logger, allowing the assessment of vibrations up to 200 Hz. The logger
Figure 2. Particle‐size distribution of mobile braided river bars, where the range is based on the May and June 2018 Wolman counts (400 particles) of surface particle size, and the sampled particle‐size distribution of bedload material in transport (truncated at 91 cm), both performed by Misset et al. (2020). A log‐“raised cosine” distribution (Tsai et al., 2012) with μ = 0.055 m and σ = 0.75 was visually fitted to the data. The photo inset shows bedload sampling from the bridge using an Elwah sampler.
was configured with an amplifier gain of 16. Simultaneously, an active GPS antenna (BY‐GPS‐07) was used to register time, allowing the precise synchronization of the measured records from the geophones at the differ-ent sites.
The raw seismic signal (counts) exported from the logger is converted to vertical ground velocity V (m/s) due to seismic ground vibrations by considering the logger and the geophone sensitivities according to manufac-turer specifications. To analyze the seismic signal in both time (t) and frequency (f) domains, the power spec-tral density P is determined as equation 1:
P fð Þ ¼jV fð Þj
2
T ; (1)
where V(f) is the Fourier transform of V(t) over a time window with duration T. We use the method of Welch (1967) to compute P, which is based on a fast Fourier transform and averages power using three over-lapping time subwindows with 50% overlap to reduce the variance in the computed spectrogram following Burtin et al. (2016). The temporal resolution T over which power is determined is 1 s for the active drop‐experiment and 3 s for bedload monitoring. For the long‐term bedload monitoring, we aggregate the signal to a 10‐min resolution by taking the median P value, which allows the effects of episodic and short‐lived, strong‐energy signals, for example, the passing by of people, animals, or vehicles, to be excluded from the time series.
3.2. Seismic Framework
Following Tsai et al. (2012), we use a general seismic framework to relate the impact force of particles on the riverbed to seismic power measured on the riverbank through seismic wave generation and propagation. The absolute vertical ground velocity V of seismic ground vibrations with wave frequency f may be described as equation 2:
V fð ; rÞ
j j ¼ 2πfF fð ÞG f ; rð ÞZ fð Þ; (2) where F is the vertical component of the impact force exerted on the riverbed, G is the Green's function that converts force to ground velocity and describes the frequency‐dependent wave attenuation with radial dis-tance r from the source, and we introduce Z as a site‐specific amplification factor that was not specified in Tsai et al. (2012) and which we will discuss later.
The impact force of a typical bedload particle may be considered as instantaneous because relative impact duration (milliseconds; Johnson, 1987) is expected to be much shorter than the considered seismic wave per-iod (tens of milliseconds). In that case, force is a dirac, that is, it is no longer a function of frequency and can be described as equation 3 (Tsai et al., 2012):
F¼ muγ; (3)
where m is the mass of the particle, u is its vertical impact velocity, andγ is a factor that accounts for the elasticity of the particle impact whereγ = 1 corresponds to a perfectly inelastic impact and γ = 2 to a perfectly elastic impact.
The Green's function for Rayleigh waves is expressed as equation 4 (Aki & Richards, 2002; Tsai & Atiganyanun, 2014): G fð ; rÞ ≈ Nk 8ρsvpvg ffiffiffiffiffiffiffiffi 2 πkr r e−vgKπfr: (4)
This function describes wave amplitude decrease due to two‐dimensional spreadingpffiffiffiffiffiffiffiffiffiffiffiffiffi2=πkrin the far‐field domain (wavelengthλ = vp/f≫ r), where k is the angular wavenumber (rad/m), equation 5a, and inelastic
attenuation of waves as they propagate into the ground. N is a function of the surface wave eigenfunction which was approximated to unity by Tsai et al. (2012) and which was later supported by numerical modeling in Tsai and Atiganyanun (2014) for frequencies of 10–50 Hz that are typically associated with bedload. The
density of sedimentρsis 2,650 kg/m3, vpis the wave phase velocity, equation 5b, vgthe wave group velocity,
equation 5c, and K is the dimensionless quality factor, which is an intrinsic material parameter that accounts for inelastic energy loss, equation 5d.
k¼ 2πf =vp; (5a)
vp¼ vp0ðf=f0Þ−ξ; (5b)
vg¼ vp=ð1 þ ξÞ; (5c)
K¼ K0ðf=f0Þη: (5d)
Tsai et al. (2012) used normalized wave velocity functions (f0 = 1 Hz) and provided standard values of
vp0 = 1295 m/s (vp at 1 Hz) and ξ = 0.374, a dimensionless exponent that is used to quantify the
frequency‐dependency decay, based on depth‐velocity profiles for a generic bedrock subsurface (Boore & Joyner, 1997). It is uncertain however to which extent such approximation applies to river settings that contain sedimentary layer(s) in the subsurface. Lastly, Tsai et al. (2012) assumed a constant K = 20 for the relatively high frequencies of interest, that is, K0= 20 (K at 1 Hz) and a dimensionless exponentη = 0,
equation 5d, based on Anderson and Hough (1984). Note that the term K is referred to as Q in Tsai et al. (2012), but that here Q is used for discharge. For the local application of the seismic framework, the functions and the values of the constituent parameter values given here have substantial uncertainties associated with them and we therefore seek to quantify them for ourfield site by means of an active seismic experiment.
3.3. Active Seismic Experiment
An active experiment is used to characterize riverbed seismic wave propagation and attenuation such that we can constrain ground velocity (V) due to an impact force (F) following Tsai et al. (2012). Although there are various advanced seismic sources and configurations (e.g., Park et al., 2007; Rix et al., 2002) that may be used to perform active seismic experiments, we chose a basic and practical approach. The experiment was performed during low‐flow conditions (mean discharge 4.5 m3
/s) and consisted of the repetitive drop-ping of a large rock on the dry riverbed, 10 drops at each site, at well‐defined distances along a cross sec-tion between geophones located on opposite banks (Figure 1). The rock had a mass of 17.4 kg and was dropped from a height of approximately 1.6 m (operator shoulder height). Based on our experiences, we assume purely inelastic impacts (γ = 1) as the rock generally rebounds very slightly of the surface and there is only minor loss in vertical energy transfer due to the rock glancing off or rotating after impacting upon the undulating surface. With this drop‐experiment, we specifically seek to quantify vg, vp, and K in
the Green's function, equation 4, and the amplification factor Z for the braided reach, equation 2, as a function of frequency.
First, a wave group velocity dispersion curve is determined based on identifying rock‐impact arrivals at geo-phones on opposite banks for various frequency intervals (e.g., Ritzwoller & Levshin, 1998). The raw seismic time series (in counts) are band‐pass filtered for 50% overlapping intervals of 6 Hz, using a first‐order filter, and the signals are bounded using a Hilbert transform envelope for which we picked the peak values. With this technique, the selected mode is a priori unknown and corresponds to the mode with the highest energy. Nevertheless, we assume that the fundamental mode dominates the energy over the investigated frequency range, which we verify a posteriori with the form of the inferred dispersion curve (absence of discontinu-ities). To ensure a significant time delay between arrivals on opposite banks (and reduce uncertainty due to the limited temporal resolution of the measurements), we use the impacts closest to one or other sensor on each bank (Figure 1) for which the registered peak is above the ambient noise level on both banks (we confirm our results with the second nearest impact sites). We relate the delays to known source‐to‐receiver distances (geophone and impact locations were measured with a differential GPS; Figure 1) in order to calculate wave group velocity.
Second, the quality factor K is determined throughfitting the decay of seismic power with distance. We cor-rect for background noise (period before and after drop events) and we normalize P to avoid any effects
related to signal amplitude and distance (see Supporting Information S2). The value of K is derived and we assess its frequency dependency (e.g., Erickson et al., 2004), that is, the value ofη (equation 5d).
Third, we quantify Z to account for potential site amplification effects that may occur due to the trapping of seismic waves in sedimentary layers or in the structure of the basin topography (e.g., Roten et al., 2004). We effectively use Z as a semi‐empirical factor to calibrate the seismic amplitude of the impacts in a field setting, and therefore, it also includes the errors associated with the assumed values of factors N andγ. Z is quanti-fied with equation 2 using the locally derived Green's function (equation 4) and the prescribed impact force that the rock exerts on the bed surface (equation 3). The force is derived from the known mass of the rock and an estimate for the impact velocity u¼ ffiffiffiffiffiffiffiffiffi2ghi
p
when it is released from height hi(g is the acceleration
due to gravity). To facilitate this simple experimental design, we (1) performed the impacts on the dry riv-erbed, not in the water, and (2) we use a large rock, relative to the surface material and relative to surface deformations upon (repetitive) impact, to minimize impact force energy losses upon impact with the uncon-solidated sedimentary surface (Bachelet et al., 2018; Kean et al., 2015). Finally, an additional small‐scale drop experiment, consisting offive drops at five distances (30–70 m), was performed at site B1 to verify local site amplification effects.
3.4. Physically‐Based Bedload Transport Model
To couple seismic measurements to bedloadflux, we require a particle‐based approach. The model used by Tsai et al. (2012) was taken from physically‐based and flume‐based studies that investigated bedload salta-tion for the applicasalta-tion in studies onfluvial bedrock incision (Lamb et al., 2008; Sklar & Dietrich, 2004). Saltation is considered the dominant mode of bedload transport, even at low transport rates (Lajeunesse et al., 2010), and is thus expected to be suitable to quantify bedloadflux. Using a particle‐based approach, the (downstream) bedloadflux can be coupled to the particle impacts perpendicular to the bed surface as equation 6a: qbð Þ ¼D mJL W ; (6a) qbð Þ ¼D mJUbhb causW ; (6b)
where qb(kg/m/s) is the fractional (particle size dependent) bedload transport per meter width W, for
sphe-rical particles with mass m =ρs(π/6)D3and where D is the particle diameter. Further, J is the particle impact
rate per unit channel length (m−1s−1) and L is the saltation hop length. Here, we revisit the transport model, staying with the essence of this formula 6a and estimating L based on recent particle tracking experiments. Tsai et al. (2012) used an expanded version of this formula with separate (semi)empirical descriptions of bed-load layer velocity (Ub), height (hb), particle ascent time (ca), and settling velocity (us) based on Sklar and
Dietrich (2004), equation 6b (we do not use this formula except where we explore the general sensitivity and uncertainty of bedload inversions). Note we also determine qbin kg/m/s, whereas Tsai et al. (2012)
con-sider volumetric bedload transport rates. Before we can quantify qbfor a given particle size, we willfirst
pre-sent aflume‐based expression for L and u, the latter which is used in the subsequent derivation of J from seismic power P measured in thefield.
As a basis to quantify physical bedload saltation parameters (e.g., hop length), we use scaling functions based on transport stage (Abbott & Francis, 1977), which is the ratio between the dimensionless shear stressτ*at the bed surface, equation 7a, and the critical valueτ*cthat is required to entrain particles with diameter D.
τ*ð Þ ¼D hS sD; (7a) τ* cð Þ ¼ τD *cD50 D D50 −0:9 ; (7b)
where h isflow depth at which bedload transport occurs, S is the riverbed slope, and s is the submerged spe-cific density of the bedload material, s = (ρs− ρw)/ρw, whereρwis 1,000 kg/m3. For the critical dimensionless
shear stress, we useτ*cD50=0.047, derived using the slope‐dependent function proposed by Lamb et al. (2008), and take into account particle size effects at the riverbed surface with respect to the median size (D50),
according to Parker (1990), equation 7b. Through equating equations 7a and 7b, we solve for water depth and discharge that is required for the onset of particle motion. In this study, we propose to quantify both hop length L and impact velocity u as a function of transport stage.
The hop length of a saltating bedload particle can be defined as equation 8: L Dð Þ ¼ aLD τ * τ* c −1 bL for τ*>τ*c; (8)
where aLand bLare dimensionless empirical constants for which Sklar and Dietrich (2004) found values of
aL= 8 and bL= 0.88 based on a compiled database of experimental studies. Auel et al. (2017a) reassessed this
database and expanded it with laterflume studies (Ancey et al., 2002; Chatanantavet et al., 2013) and their own experiments. They found smaller hop lengths aL= 2.3, while the scaling factor remained similar,
bL= 0.8. We use the latter model parameters in this study, which are also consistent with observations by
Gimbert et al. (2019); Figure S3‐1a in Supporting Information S3), and explore the potential implications of the application of other values.
Similar to hop length, the vertical impact velocity u of bedload grains can be described as equation 9:
u Dð Þ ¼ au ffiffiffiffiffiffiffiffi gsD p τ* τ* c −1 bu for τ*>τ*c: (9)
Sklar and Dietrich (2004) estimated a value for impact velocity u based on relations of hop height, trajectory (ratio rising and falling limb), and descent time, which was rewritten as a function of transport stage by Auel et al. (2017b) with constants au= 0.84 and bu= 0.18. Directflume measurements of impact velocity
distribu-tions have since then been done in supercriticalflow experiments for smooth (Auel et al., 2017b) and rough flow conditions (Gimbert et al., 2019). Gimbert et al. (2019) derived values of au= 0.51 and bu= 0.25 for
near‐vertical impacts, which we use as model parameters in this study considering the rough nature of the riverbed.
To determine fractional bedloadflux with equation 6a, we need the particle impact rate J, which we can derive from the (total) measured seismic power P and the modeled power associated with random individual impacts Pi. We determine Pithroughfilling equations 2, 3, and 4 into equation 1, resulting in equation 10:
J¼ P=Pi≈ P f ; r; Dð Þ ρs 2v
p3vg2
Z2π2f3m2u2χ
−1ð Þ:β (10)
In the implementation of equation 3, we assumeγ = 2, that is perfectly elastic impacts of particles as they travel under water over a consolidated riverbed (these conditions differ from the active drop experiment with a large rock where we assumeγ = 1), conforming to Tsai et al. (2012). Further, we use u as defined in equa-tion 9 andχ(β), which is a function that approximates the amplitude decay of ground vibrations (expressed through the exponential in the Green's function, equation 4) for an infinitely long and straight river observed at a minimum distance r in equations 11a and 11b (Tsai et al., 2012):
χ βð Þ ≈ 2ln 1 þ β −1e−2βþ 1 − e −βe−βqffiffiffiffiffiffiffiffiffiffiffiffiffi2πβ−1; (11a)
β ¼2πr 1 þ ξð Þf1þξ−η K0vp0f0ξ−η
Finally, to determine bedloadflux over a complete particle‐size distribution (p(D); Figure 2), we integrate the fractional loads from equation 6a as follows in equation 12:
qb¼ ∫ Dmax Dminp D; τ* τ* c qbð ÞdD:D (12)
We truncate and (re‐)normalize the particle‐size distribution to consider only those particles that are in motion (τ*>τ*
c), thus accounting for partial transport conditions.
3.5. Model Application for Bedload Quantification
3.5.1. Scaling Analysis
Ourfirst assessment of the seismic bedload model is based on a scaling approach where we assess the power‐based dependency of seismic power on sampled bedload flux. This gives insight into the coherence of the seismic response (e.g., Burtin et al., 2011) and the contributions of the different sources to the total seismic power (e.g., Roth et al., 2016; Roth et al., 2017). According to Tsai et al. (2012), seismic power due to bedload transport is expected to scale to the power 2 with particle mass, Pb~m2(equation 10), to scale
lin-early with bedloadflux given a constant particle size, Pb~qbD943(equations 6a and 10) and to scale to the
power 6 with particle diameter, Pb~m2~D946(equations 6a and 10), since m~D3and the dominant particle
size is the 94th percentile (D94). If we include the effect of particle‐size dependency for hop length (equation
7) and impact velocity (equation 8) on bedload flux (equation 6a), the scaling relation becomes Pb~qbD943.2~D946.5. To determine scaling relations, wefit the data using major axis regression which takes
into account the uncertainties in both the inverted and sampled bedloadfluxes. Fitting is performed on the basis of the perpendicular distance of each data point to the regression line rather than the distance along the y‐axis as is done in ordinary least squares regression.
In addition, we perform a scaling analysis using the measuredflow discharge, where seismic power due to flow turbulence (Pw) is expected to scale with discharge as Pw~Qw1.4, for channels where W≫ h (Gimbert
et al., 2016). This scaling is expected to be manifest at frequencies lower than those characterized by bedload transport (Gimbert et al., 2014). For the scaling with discharge, we use nighttime values, from 22:00 to 06:00, to limit potential effects of anthropogenic background noise (McNamara & Buland, 2004). Indirectly, we can then also address seismic power due to bedload transport (Pb) following the bedload rating curve defined for
this site by Misset et al. (2020), qb~Qw5.3. Because no systematic change in particle size has been observed
during the sampling period (Misset et al., 2020), seismic power may be expected to scale linearly with bed-loadflux and therefore Pb~Qw5.3.
3.5.2. Bedload Flux Inversion
For the inversion of bedloadfluxes, we use the locally derived seismic characterization (section 3.3) and revised formulation of the physically‐based bedload transport model (section 3.4; equation 6a). For the latter, we calculate bedload dynamics for the particle‐size distribution of the riverbed as determined for the braided reach (Figure 2). Misset et al. (2020) have shown that this reach is an important source of sediment that is exported downstream where similar particle‐size distributions were sampled and where we invert bedload transport rates (B1; Figure 1). We visuallyfitted a log‐“raised cosine” distribution, as introduced by Tsai et al. (2012), to the sampled data sets, taking particular care to get an accurate representation of the largest fractions that contribute strongly to the seismic signal (Figure 2). The distribution was discretized with par-ticle fractions dD of 5 mm, ranging from Dmin= 1.6 mm to Dmax= 0.44 m to determine the total load with
equation 12. For the particle dynamics, we determine hop length and impact velocity based values forτ*and τ*
cfor particles in motion observed at B1, given that the criterion for motion is exceeded in the braided reach
source area (Figure 1). This calculation was however conditioned by sediment mobility in the upstream braided reach, where we use the values derived by Misset et al. (2020); appendix A therein.
We compare the inverted bedloadfluxes with sampled fluxes derived from Misset et al. (2020). Although we use the sampled loads for the validation of the seismic approach, we would like to emphasize here that direct bedload measurements have a considerable uncertainty (Vericat et al., 2006). For this reason, we employ major axis regression as mentioned earlier. Considering the assumptions and uncertainties associated with both the seismic framework and bedload dynamics, we use a number of scenarios to explore the implications of different plausible parameter sets on inverted bedloadflux.
4. Results
4.1. Experimental Characterization of Riverbed Seismic Response 4.1.1. Seismic Wave Propagation
The seismic waves generated by individual rock impacts on one bank were observed by a geophone on the opposite bank at a distance of more than 100 m (see example Figure 3a). Frequencies for which the impacts could be identified lie in the range of 10–30 Hz. The determined wave group velocities differ little between the upstream and downstream cross section, apart from outliers (symbols without afill) that are associated with higher‐order modes and that we do not consider here. The data are described with a power‐law dependency on frequency (Figure 3b), least squaresfitting of the function vg= vp0(f/f0)−ξ/(1+ξ) that
fol-lows from combining equations 5b and 5c. The observed velocities are within the range (120–920 m/s) found by Huang et al. (2007), using a similar experiment in debrisflow channels, but are considerably lower than the reference values provided in Tsai et al. (2012) for generic bedrock sites. We also note that from ~16 Hz, velocities appear to approach a constant value of ~180 m/s, which can be related to the pre-sence of a distinct sedimentary top layer in the subsurface (on the order of one wavelength or ~10 m thick) that has relatively uniform wave propagation. Unlike wave group velocity, wave phase velocity cannot be identified visually and we calculate these with equation 5c using the values vp0= 680 m/s andξ = −0.31
obtained from Figure 3b.
4.1.2. Attenuation of Seismic Power
Spectrograms of the active drop‐experiment show that seismic power associated with rock impacts decays with distance (Figure 4a). At a short range, impacts may be observed over a wide frequency band of 10–70 Hz, but this range becomes narrower, ~15–40 Hz at a distance of 150 m for site R2 (Figure 4b; for spectrograms of sites L1, L2, and R1, see Supporting Information S1). The lower half of this frequency range corresponds to the range for which wave group velocity could be determined. Frequencies around 10 Hz have a higher level of background noise, which is likely to be generated by riverflow turbulence (Burtin et al., 2011; Gimbert et al., 2014).
Using thefitted velocity function from Figure 3b and limiting the extrapolated values for high frequencies to 180 m/s (Figure 3b), we derived fitted values for the quality factor K (Figure S2‐1 in Supporting Information S2). The response of seismic power as a function of distance from impact and frequency is well described using the Green's function (equation 4) as shown in Figure 5. Scatter is smallest for the higher frequencies at which seismic power decreases more rapidly with distance. The observed frequency dependency of K (fitted with equation 5d) is not very strong, which is consistent with the presumed pre-sence of a uniform subsoil. The value approaches 20 (Figure 5 inset), which was used as a constant by Tsai et al. (2012).
Figure 3. (a) Seismogramfiltered for frequency (19.5–25.5 Hz in this example) bounded with a signal envelope and identified peak amplitudes corresponding to 10 drops that were performed at 17 m from L2 and 121 m from R2; the inset provides a zoom‐in of the first drop. (b) Wave group velocity per filtered frequency range (50% overlapping bins of 6 Hz) between geophones on opposite banks; for example, L1➔ R2 indicates the velocity of waves that originate from the impact site closest to L1. Each data point represents the median of 10 drops; the size gives an indication of the confidence based on the standard deviation of the registered time delay between these drops; larger data points indicate a lower standard deviation and vice versa. A regression curve (black) that describes the wave group velocity dispersion curve based on equations 5b and 5c wasfitted using all the data, excluding the outliers, shown with no fill, that are associated with higher‐order modes.
4.1.3. Seismic Inversion for Large‐Rock Impact Experiment
To validate the seismic framework, we invert the mass of the rock used in the active drop‐experiment, multi-plied with the residual amplitude factor Z, from the seismic measurements using equations 2 and 3. For the frequency range 12.5–37.5 Hz, in which impacts could be identified, the mass of the rock is well estimated with no evidence of net effects on the seismic amplitude (Z = 1; Figure 6). Residual standard deviation among the observations is approximately ±5 kg or ±30% of the known mass. When applying the wave pro-pagation and attenuation parameters suggested by Tsai et al. (2012) as afirst estimation, the seismic inver-sion also gives reasonable values, although there is a frequency‐dependent bias in the inverted mass. For lower frequencies, the application of these parameters may lead to an overestimation of a factor 2–3, but at higher frequencies, this error is marginal.
These results not only demonstrate that the seismic framework is applicable in a gravel‐bed river setting but also highlight the need to locally derive the constituent parameters of the Green's function. We have shown that this may be accomplished with a simple active experiment. To be able to invert bedloadfluxes at site B1, we used an additional small‐scale drop experiment to verify the application of the calibrated Green's func-tion and the presence of any local site amplification at B1. The resulting Z values (Figure 6 inset) indicate that the seismic framework can be applied quantitatively, with the note that at 12.5 Hz there is an apparent site amplification effect.
4.2. Continuous Bedload Quantification From
Seismic Measurements
4.2.1. Seismic Power Scaling Relationships
Seismic power shows frequency‐based power‐law scaling with bedload transport (Figure 7a; exponent is the slope in log‐space). At 50 Hz, the scaling is approximately linear (exponent≈ 1), which is consistent with predictions from Tsai et al. (2012) for bedload‐induced seismic vibrations, assuming a constant particle size distribution. Bedload still appears to dominate the signal at 30 Hz, as the scaling exponent is only slightly lower than 1, but this is no longer the case at 10 Hz where the exponent of the power‐law is much smaller. Note that the use of major‐axis regression, which also accounts for uncertainties in the bedload sampling, leads to considerably higher regression slopes than ordinary least squares regres-sion (exponents of 0.24, 0.71, and 0.78 for 10, 30, and 50 Hz, respectively). The data appear to show an increase in scatter with increased seismic power and there is a potentialflattening‐off in the data of 30 and 50 Hz from approximately 10log10qb=−8 (the 10 largest sampled qbvalues fall
below the regression line).
Figure 4. (a) Spectrograms of a series of 10 impacts that were performed at 15 m from R2 (top) and 124 m from L2 (bottom). (b) eismic power with distance from rock impact observed with geophone R2; the spectrograms at the time of the impact in panel (a) are represented as a line in panel (b).
Figure 5. Seismic power decay with distance per frequency following the expression of the Green's function (equation 4). The inset shows the quality factor K derived fromfitting normalized power based on the Green's function (see Supporting Information S2).
When we account for sample‐dependent particle size, the scatter around the scaling relation with bedload transport improves in terms of an increase in Spearman's r value (we use Spearman's rank correlation because of the nonlinear axes) and a decrease in the median residual from a factor 1.2±1to 1.1±1for 30 and 50 Hz (Figure 7b). Despite uncertainties associated with the small data set (for this reason, wefixed the exponent of the regression based on Figure 7a) and limited range in particle size that is caught during sampling, this result indicates that particle size dynamics may account for a significant part of the residual seismic variability in Figure 7a and supports the physical basis of the applied bedload transport model. Of Figure 6. Inverted mass multiplied with the net amplification factor (Z) plotted against frequency for the active drop‐experiment. The known rock mass is shown assuming Z = 1, while the dashed line is derived based onfirst‐estimate parameters in Tsai et al. (2012). The boxes give the range of data per frequency that lies between the 25th and 75th percentile within the boxes horizontal lines that mark the median value; the whiskers covers the range without outliers (0.7 and 99.3 percentile), and the outliers are marked as +. The inset shows the Z values per frequency for the braided river reach and at site B1. Here, we also derived Z values that would be required to obtain the prescribed mass using the set of Green's function parameters given in Tsai et al. (2012).
Figure 7. (a) Seismic power for 10, 30, and 50 Hz against bedloadflux scaled as a 10log10value to match the dB scale of the seismic power; major axis regression
lines describe the power relations per frequency. (b) Seismic power for 10, 30, and 50 Hz against bedloadflux multiplied by D943.2(10log10scaling) for the
samples for which particle size was determined (13 of the 51 samples in panel (a) for which seismic measurements were available). For these samples, the values are plotted using both the mean particle size and the sample‐specific measured particle size which leads to increased Spearman's r values (shown in the legend). The major axis regression line isfit to the data, given the exponents derived in panel (a), and offsets of a factor 1.1±1perpendicular from these
Figure 8. Plots of seismic power per frequency 10 Hz (a), 30 Hz (b), and 50 Hz (c) against discharge scaled as a 10log10
value to match the dB scale of the seismic power. Data are shown for the whole season (color scale indicating months). Least squares regression lines describe the seismic power for discharge above and below the modeled threshold of full particle mobility,τ*=τ*cðDmaxÞ > 1 at Qw=11.9 m3/s, where there is an apparent change between a turbulence‐
dominated and bedload‐dominated seismic signal. For panel (c), the line below threshold is dashed due to poor fit, and we added two additional lines where P~Qw5.3as a reference to indicate different phases with ample sediment supply
course, strong deviations may still be expected due methodological limitations to resolve the temporal and spatial variability in bedloadflux.
We can indirectly verify the bedload sample scaling relationships with those obtained with discharge (Figure 8). At a low frequency of 10 Hz, the power‐law scaling can be described with a continuous func-tion with an exponent of 1.5 (Figure 8a; note we use least squares regression here as the uncertainty in discharge is negligible) that is very close to the value of 1.4 expected based on the physical modeling of turbulentflow (Gimbert et al., 2016). It does not mean that no bedload transport occurs, but rather that at such a low frequency, bedload‐induced ground vibrations are overwhelmed by vibrations due to flow turbulence. Because flow turbulence drives bedload transport, we find a clear scaling relation with sampled rates at 10 Hz in Figure 7 but, due to the nonlinear nature of bedload transport with discharge, with a lower exponent value.
At 30 and 50 Hz, a clear change in scaling power occurs as discharge exceeds a value (11.9 m3/s) that cor-responds to the modeled threshold of full particle mobility (Figure 8b, 8c) and can be related to pavement breakup in the braided reach as observed here by Misset et al. (2019). For discharges higher than this cri-tical value, the power‐law trend is steeper with an exponent of 4.4 at 30 Hz and 5.3 for 50 Hz. The latter corresponds to the qb~Qw5.3rating curve found by Misset et al. (2020) under the condition that Pw~qbas
predicted by Tsai et al. (2012). The trend also describes a more scattered data set that exhibits a clockwise hysteresis in May–July (note that the hysteresis may also be detected at 10 Hz in Figure 8a although it is clearly less pronounced). The wider scatter is similar to the plot with sampled load (Figure 7a), but in this case, there is no evidence of signal saturation. For discharges lower than the identified threshold, mainly in August–September when little to no bedload transport was observed, the trend is similar to that for 10 Hz, although slightly lower exponents are found. For 50 Hz, the lower discharges (<11.9 m3/s) now exhibit a baseline response, which still has a power scaling of just more than 1, but also shows a period with a steeper response (indicated by a thin red line with exponent 5.3). The latter follows directly after a largeflood on 9 August (peak flow ~45 m3/s), which temporarily led to low rates of partial transport that are not visible at lower frequencies due to the dominant effect offlow turbulence (note the differences in the y‐axes limits in Figures 8a to 8c). This period of transport may be associated with changes in channel morphology and sediment availability, that is, the reworking of channel‐lag deposits after the flood. Periods of increased sediment supply and associated changes in critical discharge may therefore be observed, while the characteristic bedload scaling relationship remains constant.
The results of the scaling analysis are summarized in Figure 9, showing both the exponent (a) and the Spearman's r value of the relationship (b). For the lower discharges, we see a clear scaling of 1.4 with dis-charge (r > 0.8) for frequencies up to ~15 Hz, which is in agreement with Gimbert et al. (2014). At higher frequencies, this scaling weakens as bedload transport becomes dominant as indicated by the exponent (Figure 9a). Due to hysteresis, the frequency range for bedload cannot be solely identified on the basis of the r value (little variation with frequency in Figure 9b), as is the case for the frequency range for turbulence. The scaling of seismic power against discharges for which full particle mobility is expected on one hand and the scaling of seismic power against sampled bedloadflux on the other hand are remarkably similar and in agreement with Tsai et al. (2012). They both increase between 15 and 30 Hz, indicating a transition from a flow turbulence‐dominated to a bedload‐dominated seismic signal. Although the discharge relation is slightly less strong, it provides a good control and potential alternative for the bedloadflux scaling. The range of frequencies for which bedload may be inverted is 30–70 Hz, above which the exponents of the scaling rela-tions (Qw>11.9 and qb) rapidly decrease and become much smaller than the expected values (1 and 5.3,
respectively) for bedload transport.
4.2.2. Bedload Flux Inversion From the Seismic Record
When applying the physically‐based bedload model, using locally derived seismic, hydraulic, and particle size parameters, we invert transport rates from measured seismic power for different frequencies (Figure 10). The results show that the seismically derived values for frequencies in the range of ~30–50 Hz correspond well to those derived from instream sampling (qb(P)≈ qb) across the full range of transport rates
from less than 0.01 to nearly 1 kg/m/s. The data scale nearly linearly against each other (as seen earlier in Figure 7a assuming a single particle size and not a discharge‐dependent distribution) and the magnitude of the inverted values is also in good agreement with the independently obtained bedload samples. Most
instantaneous values fall within a factor of 5±1(i.e., between 0.2 and 5), while the regression lines lie within a factor of ~2±1; the regression line of 50 Hz shows negligible error. Only for 25 Hz, there is markedly lower exponent and general overprediction of bedload rates, which may be directly attributed toflow turbulence that significantly affects the seismic signal at these low frequencies (Figure 9a). Although seismic power scales with bedloadflux up to frequencies of 70 Hz (Figure 9a), we limit the quantitative inversion to 50 Hz due to the uncertainties associated with the extrapolation of the results of the seismic characterization (Figure 6).
4.2.3. Modeling Sensitivity and Uncertainty
Although we have applied the modeling to local conditions, there remain uncertainties in the seismic char-acterization (wave propagation and attenuation) and assumptions concerning (modeled) bedload dynamics. These uncertainties may propagate into the inverted bedloadfluxes as illustrated for a number of plausible scenarios in Table 1.
In Scenario 1, we use thefirst‐estimate seismic parameter values for vg, vp, and K that were provided by Tsai
et al. (2012) to quantify the relevance of the active drop‐experiment on bedload fluxes. The derived bedload rates are systematically higher than those from the applied model with the largest differences occurring at low frequencies, a factor of nearly 4 at 31.25 Hz (2.03 vs 0.55 for factor a in Table 1). Note that the relative difference in inverted transport rates is larger than that of the inverted masses of a single rock at this fre-quency (factor ~1.5 in Figure 6). This is because, for a given seismic power, the apparent error in the active experiment in terms of Z−1scales linearly with force (equation 2) and thus mass (equation 3), while in the Figure 9. Characterization of seismic power per frequency for discharges above and below Qw=11.9 m3/s, the estimated
threshold of full particle mobility. (a) Power of the scaling relation, slope of P/Qwand P/qb(discharge on the left axis
and bedloadflux on the right axis) on a log–log scale as in Figures 7 and 8. Dashed lines indicate predicted scaling values for bedload and turbulentflow as predicted based on the physically‐based models of Tsai et al. (2012) and Gimbert et al. (2014) respectively. (b) Spearman's r values associated with the scaling relations. Thin vertical lines in panels (a) and (b) indicate the frequencies 10, 30, and 50 Hz.
case of bedloadflux, for a given power and particle size, Z−2scales with impact rate (equation 10) and bedloadflux (equation 6a). The nonlinear propagation of Z makes it important to obtain a good field constraint on its (effective) value.
In Scenario 2 (Table 1), we use the detailed description of bedload dynamics following Tsai et al. (2012) while using the seismic parameters as derived in this study. The results vary slightly from those of Scenario 3, where we simplified the bedload dynamics through assessing hop length and impact velocity directly based on transport stage (see Figure S3‐1 in Supporting Information S3 for a comparison of the L and u values) even through the underlying database on which the empirical formulations are based is the same. For both Scenarios 2 and 3, the exponent b of the regression is larger than in the applied model (0.94 and 0.95 vs. 0.90; Table 1). In Scenario 4, we use the values derived for L and u by Auel et al. (2017a, 2017b) forflow conditions over afixed planar bed, which leads to much higher (order of magnitude) estimates of bedload transport due Figure 10. Double‐logarithmic plots of bedload transport rates derived from seismic power qb(P) using the physically‐based model compared to sampled loads qb
per frequency. The power‐law relations were derived using major‐axis regression; lines indicating qb(P) =qband offsets of a factor 5±1perpendicular from the
regression line are provided as a reference. Note that not all data points are shown for 25 Hz (the limits of the y‐axes are the same for all plots) and that the regression line of 50 Hz lies more or less on the line qb(P) =qb.
to a reduced impact velocity (Figure S3–1). In Scenarios 5 and 6, uncertainties regarding the value of the dimensionless critical shear stress (e.g., Buffington & Montgomery, 1997; Recking, 2009) are explored through the use of respectively lower and higher values that may occur in this type of setting (in the applied model τ*cD50 = 0.047), which may have a large impact on the exponent of the regression due to changes in the particle‐size distribution that is in motion. In Scenario 6, the particle distribution effectively becomesfiner at low discharges due to the increased critical shear stress, resulting in a higher bedloadflux to account for the generated ground vibrations. A change in particle‐size distribution may occur due to upstream changes in sediment sources (e.g., sediment delivery from a debrisflow). A general fining of −1 cm in Scenario 7 will lead to a large overestimation of the bedload flux of ~2 orders of magnitude. Conversely, in Scenario 8, the coarsening of the entire particle‐size distribution (+1 cm) leads to an underestimation of the bedload flux (order of magnitude) that may account for the generated ground vibrations. Thesefindings illustrate that the particle‐size distribution is a key factor in the applied model (Tsai et al., 2012) and needs to be well constrained in thefield.
5. Discussion
5.1. Constraining Seismic Parameters From Rock Impact Force
In this study, we consider the application of a seismic framework through performing a simple active drop‐experiment to characterize seismic wave propagation and attenuation in a braided riverbed. Using geo-phones on opposite riverbanks, we quantified wave group velocity, ~200 m/s at 25 Hz, which is commensu-rate with values found in debrisflow channels (Huang et al., 2007). The velocities are however markedly lower than those in generic bedrock settings (Boore & Joyner, 1997) which have been used as afirst estima-tion in Tsai et al. (2012). Rayleigh waves appear to exhibit little dispersion in this riverbed setting at the fre-quencies of interest (>20 Hz; Figure 3b) and it may therefore be justified to assume a constant velocity due to the wave propagation through a relatively shallow and uniform sediment layer. Using a Green's function, we could approximate the frequency‐dependent decay of wave amplitude with distance and derive the quality factor K of the subsurface, which is close to 20 (Figure 5a) and in line with the value used by Tsai et al. (2012). Most importantly, we show that the relatively simple experimental design and the (semi‐)empiric nature of the seismic framework can adequately describe both seismic wave propagation (Figure 3b) and wave attenuation in the riverbed (Figure 5b). The difference in the spatial decay of seismic power at frequencies dominated byflow turbulence and bedload transport, as dictated by the Green's function parameters, may be used to determine a suitable distance from the river to monitor bedload (see Gimbert et al., 2014). Although the obtained wave velocity and quality factor K may be indicative for this type of gravel‐bed river setting, we strongly recommend local derivation of the interdependent seismic parameters (Morozov, 2010) that may vary to a large extent due to the spatially variable nature of the subsurface (e.g., Huang et al., 2007). In afield setting, it is critical to assess the absolute amplitude of the seismic signal as any residual error will affect inverted bedload transport rates in a nonlinear fashion (scenario 1 and 2; Table 1). Based on the active Table 1
Sensitivity Analysis of Bedload Inversion for 31.25 and 50 Hz Based On the Factor a and Exponent b of the Regression Function qb(P)≈ aqbb
Frequency (Hz) scenario
Factor a: qb(P)≈ a qbb Exponent b: qb(P)≈ a qbb
31.25 50 31.25 50
Applied model in Figure 10 0.55 1.10 0.90 1.04
1. vg, vp, and K from Tsai et al. (2012) 2.03 1.80 0.90 1.04
2. Bedload dynamics from Tsai et al. (2012) 0.59 1.18 0.94 1.07
3. L and u based on S&D (2004) 0.71 1.42 0.95 1.08
4. L and u based on Auel et al. (2017a, 2017b) 14.38 28.54 0.85 0.98
5. Critical shear stressτ*c50=0.03 0.70 1.39 0.88 1.02
6. Critical shear stressτ*c50=0.065 0.31 0.60 0.65 0.76
7. Particle size D‐1 cm 101.78 202.14 0.89 1.02
8. Particle size D+1 cm 0.14 0.28 0.91 1.05
Note. The parameterization of the applied model is described in section 3, and the values of a and b correspond with the results in Figure 10. For the alternative Scenarios 1–8, see the main text. S&D (2004) is a shorthand reference to Sklar and Dietrich (2004).
experiment across the braided reach, we found no significant amplification effect (Figure 6) and we could accurately invert the mass of the rock used in the active drop‐experiment from seismic measurements for a range of frequencies (~15–40 Hz) with a standard deviation of approximately ±5 kg or ±30%. The experi-ment and analyses we present may be a simple yet important means to quantify seismic response and to con-strain the seismic power associated with bedload transport.
5.2. Quantifying Bedload Flux Using a Physically‐Based Bedload Model
In this study, we show the presence of a pronounced bedload signal in seismic power records that scales nearly linearly with bedloadflux in the range of ~30–50 Hz (Figure 7a, Figure 9), which is in agreement with the theoretical model of Tsai et al. (2012). Particle size variability may account for significant scatter in this scaling relationship (Figure 7b), supporting the physical basis of the bedload model we used and, at the same time, showing its sensitivity to this parameter. To complement punctual bedload measurements, we show that a scaling approach of seismic power with continuous discharge measurements allows the frequency range dominated by bedload to be clearly constrained (Figure 8), which is crucial for the accuracy of subse-quent bedload transport inversions. We confirm the applicability of the power scaling from the theoretical model of Gimbert et al. (2014, 2016), Pw~Qw1.4for 5–15 Hz, and use this together with an established bedload
rating curve (1) to verify the linear bedload scaling on a continuous scale, which allows the inversion of vari-able loads and giving confidence in the extrapolation to higher loads and (2) to assess the onset and magni-tude of bedload transport (Figure 8) which vary with temporal changes in the transport and supply dynamics associated with morphological changes in the upstream braided reach (Misset et al., 2020). The scaling ana-lysis allows the underlying processes to be distinguished based on causality, something that may not be pos-sible with statistical techniques alone. In particular, the highest r value in Figure 9b for qbis found at 15 Hz,
which is dominated byflow turbulence rather than bedload. This illustrates that empirical fitting and opti-mization approaches (e.g., Dietze et al., 2019; Roth et al., 2016) require careful application as they are suscep-tible to error and misunderstanding. The scaling analysis we present here will help to further investigate bedload processes that have been inferred from event to seasonal‐based hysteresis patterns (Barrière, Oth, et al., 2015; Burtin et al., 2008; Hsu et al., 2011; Roth et al., 2014,).
Through considering and quantifying seismic attenuation and potential amplification effects (Figure 6) in the subsurface, we could accurately invert bedloadfluxes from measured seismic power. The values corre-spond to independently sampled bedloadfluxes, with uncertainties generally within a factor of 5±1, while the measurements themselves span 2–3 orders of magnitude (Figure 10). This is not much more than the fac-tor 2±1derived in a controlledflume setting by Gimbert et al. (2019). Our results are encouraging considering the inherent uncertainties associated with both seismic and sampling techniques (Tsai et al., 2012; Vericat et al., 2006) and the natural variability in bedload transport (e.g., Gomez et al., 1989) and associated particle size (Figure 7b; Misset et al., 2020). When temporalfluctuations are averaged out, the continuous seismically derived bedloadfluxes may be expected to have an uncertainty of ~2±1(deviation regression lines in Figure 10). Based on scenario calculations (Table 1), we illustrate the sensitivity and uncertainty of the bed-load inversion results which we cannot directly evaluate with sampled bed-loads, stressing the importance of local seismic characterization and a physically‐based modeling approach.
Our modeling approach, using particle hop length and impact rate as a function of transport stage, is simpler and may be expected to be more accurate than the approach used in Tsai et al. (2012) since no prior assump-tions are made on the detailed particle dynamics, that is, the values of parameters Ub, hb, ca, and us. The
resulting differences in parameter values (Figure S3‐1 in Supporting Information S3) and inverted bedload fluxes (Table 1) are limited. More importantly, we show that relations for hop length and particularly impact velocity (Figure S3‐1) may differ with hydraulic morphological conditions, leading to deviations in inverted bedloadfluxes of more than an order of magnitude (Scenario 4; Table 1). Using the more generic model, we profit from recent additions to existing data sets which have led to revised empirical relationships such as for hop length (Auel et al., 2017a) and new measurement methods, for example, smart rocks (Gimbert et al., 2019). The latter allows impact velocity to be measured directly rather than being estimated indirectly. The use of these parameters entails an inevitable uncertainty together with uncertainties due to assumptions regarding the mechanics of bedload transport, for example, the uniform saltation, not rolling or sliding, of particles (Tsai et al., 2012), which may not necessarily be fully satisfied. Our results indicate, however, that
the net effect of these uncertainties has a relatively limited effect on the precision and accuracy of inverted bedload rates.
To successfully monitor bedload dynamics with seismic techniques, it is most important to have a good con-straint on particle size, that is, which particles are available and which particles are in transport? This is cri-tical as the (natural) variability in bedload particle size that is supplied and transported may translate to significant errors in bedload rates, particularly on short time scales. Nonlinear effects of the threshold of motion and supply‐driven changes in particle size distribution may even lead to systematic, order(s) of mag-nitude errors in bedloadflux (Scenarios 5–8; Table 1). In an attempt to overcome this uncertainty, particle size dynamics may be constrained through the choice of a suitablefield setting. First, a measurement site should ideally have a well‐known source of (coarse) bedload material; in our case, material is released from the upstream braided reach (Figure 2). Second, supply‐limited conditions at a monitoring site typically allow for a constant distribution of fully mobile, saltating particles that are independent of discharge as shown by Misset et al. (2020) in our case. These conditions may also limit potential energy dissipation effects at the bed surface due to unconsolidated bed material (Bachelet et al., 2018; Schmandt et al., 2017; Turowski & Bloem, 2016). Third, seismic monitoring at measurement sites that are instrumented with impact plates or acoustic sensors may provide information on temporal particle size dynamics (e.g., Barrière, Krein, et al., 2015; Mao et al., 2016; Wyss et al., 2016). More generally, an integrated approach where seismic mea-surements are combined with complementary (direct or indirect) monitoring techniques will provide further insight in quantitative bedload dynamics and how they are reflected in the measurements (e.g., Habersack et al., 2017; Misset et al., 2020). For longer‐term (seasonal to decadal) investigations into sediment fluxes and the mechanisms that drive these influvial and Alpine environments (e.g., river management or climate change), well‐constrained seismic measurements can be readily applied to quantify order of magnitude bed-load transport where complementary monitoring is not required or not possible due to limitations of instream techniques.
6. Conclusions
To the best of our knowledge, we present here thefirst comprehensive field validation of the physically‐based bedload transport model of Tsai et al. (2012) with independently sampled measurements. A two‐step approach is used to evaluate the applicability of various model assumptions, (semi‐)empiric relations, and proposed parameter values that underlie (1) the seismic framework and (2) the physically‐based bedload saltation modeling.
First, we show that a simple but adequate experiment with a series of large‐rock impacts allows the charac-terization of riverbed surface wave propagation and attenuation using seismic measurements from bank‐side geophones. Based on this characterization, rock mass and its impact force can be accurately inverted from seismic power with a standard deviation of approximately ±30% for the frequency range 15–40 Hz. Using first estimate parameters from Tsai et al. (2012), there is a residual frequency‐based bias in the inverted mass of a factor of ~1.5, but perhaps more importantly, the presence of potential site‐amplification effects can be verified which may lead to larger errors. Our results confirm the suitability of the seismic framework, that is, a Green's function with locally derived parameters, which is a prerequisite for the quantitative seismic monitoring of bedload transport but also other processes such as subglacialflow (Gimbert et al., 2016) or debrisflows (Kean et al., 2015; Lai et al., 2018).
Second, we clearly distinguish turbulentflow and bedload transport sources in seismic measurements using scaling relationships based on the physical models of Gimbert et al. (2014) and Tsai et al. (2012). This provides insights into the transport and supply dynamics associated with braided riverbed morphological changes and allows the bedload signal to be constrained for quantitative inversion. Using a revised formulation of the bed-load saltation model proposed by Tsai et al. (2012), we inverted continuous bedbed-loadfluxes over a range of trans-port rates (between 0.01 and 1 kg/m/s), which are in good agreement with estimatedfluxes from bedload sampling. The uncertainties are generally within a factor of 5±1for instantaneous measurements but are expected to be even smaller, ~2±1, for longer duration continuous measurements, used in sediment budget stu-dies, for example. To remain within these uncertainties, it is crucial to (1) isolate bedload‐generated seismic power from that generated byflow turbulence, or for that matter any other noise source, through discharge scaling analysis and/or the modeling of turbulence effects; (2) to constrain the available and mobile
particle‐size distribution through sampling bedload and sediment sources, the latter which requires insight into system‐based sediment storage and transfer; and (3) to consider particle dynamics, hop length, and parti-cularly impact velocity that depend on reach hydromorphological conditions.
In this study, we demonstrate that a relatively simple seismic approach can be used to accurately quantify bedload transport rates in gravel‐bed streams. Active experiments and seismic power scaling with discharge are indispensable techniques that complement instream calibration measurements (e.g., sampling), making the latter no longer strictly necessary, for example, in inaccessible or hazardous sites. Continuous seismic measurements may be readily used to investigate bedload dynamics, its interaction with riverbed morpho-logical changes, and their combined impact on bedload transfer in Alpine streams.
Notation
aL dimensionless factor used to quantify particle hop length based on transport stage
au dimensionless factor used to quantify particle impact velocity based on transport stage
bL dimensionless power used to quantify particle hop length based on transport stage
bu dimensionless power used to quantify particle impact velocity based on transport stage
ca dimensionless factor that accounts for the ascent time during particle saltation; ca= 0.66, see Tsai
et al. (2012)
D particle size diameter (m) f wave frequency (Hz) f0 wave frequency of 1 Hz
F impact force (kgm/s2)
g acceleration due to gravity; g = 9.81 m/s2
G displacement Green's function that describes the frequency and distance‐dependent attenuation of power (s2/kg)
h flow depth (m)
hb mean height of the bedload layer (m), see Tsai et al. (2012)
hi height above the surface from which a particle is released before impact (m)
J particle impact rate per unit channel length (m−1s−1) k angular wavenumber (rad/m)
K dimensionless quality factor that describes the nonelastic transfer of energy L saltation hop length (m)
m particle mass (kg)
N dimensionless amplitude coefficient in the Green's function pD probability of particle fraction dD (m−1)
P Seismic power ([m/s]2/Hz). For definition, see equation (1). In figures, values are often given in decibel (dB) relative to [m/s]2/Hz (log‐scale)
Pb seismic power due to bedload transport ([m/s]2/Hz)
Pi modeled seismic power associated with random individual impacts ([m/s]2/Hz)
Pw seismic power due toflow turbulence ([m/s]2/Hz)
qb bedloadflux per meter width (kg/m/s)
Qw flow discharge (m3/s)
r radial distance from the impact source (m)
s submerged specific density of the bedload material; s = 1.65 S riverbed slope (m/m)
t time (s)
T time window Fourier transform and temporal resolution of P u particle impact velocity (m/s)
us mean particle settling velocity through the bedload layer (m/s), see Tsai et al. (2012)
U flow velocity (m/s)
Ub mean particle velocity parallel to the bed surface (m/s), see Tsai et al. (2012)
vg wave group velocity (m/s)
vp wave phase velocity (m/s)
